Theorem f1omo 49555

Description: There is at most one element in the function value of a constant function whose output is 1_o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 49554 assuming ax-un 7733 (see f1omoALT 49557). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)

Hypothesis

Ref	Expression
f1omo.1	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omo	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo

Step	Hyp	Ref	Expression
1		1oex 8462	. . . 4 ⊢ 1o ∈ V
2		eqid 2769	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 49553	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 49476	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		df1o2 8459	. . . . . 6 ⊢ 1o = {∅}
6	5	eqeq2i 2782	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o ↔ ((𝐴 × {1o})‘𝑋) = {∅})
7		mosn 49475	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = {∅} → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
8	6, 7	sylbi 220	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
9	4, 8	jaoi 870	. . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
10	3, 9	ax-mp 5	. 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
11		f1omo.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
12	11	fveq1d 6884	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
13	12	eleq2d 2855	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
14	13	mobidv 2583	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
15	10, 14	mpbiri 261	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571  ∅c0 4294  {csn 4594   × cxp 5660  ‘cfv 6537  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1o 8452

This theorem is referenced by:  indthinc  50124  prsthinc  50126